Almost complex manifold, Almost complex manifold - Differential topology of almost complex manifolds, Almost complex manifold - Examples, Almost complex manifold - Formal definition, Almost complex manifold - Integrable almost complex structures, symplectic manifold, Kähler manifold, Chern class
ARTICLES RELATED TO Almost complex manifold - Formal definition
Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of rank (1,1) such that J2 = −1 when regarded as a vector bundle isomorphism J : TM → TM on the tangent bundle. A manifold equipped with an almost c ...
Every complex manifold is itself an almost complex manifold. In local holomorphic coordinates zμ = xμ + iyμ one can define the maps
or
One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be induced by the complex structure, and the comp ...
Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V-, so an almost complex structure on M allows a decomposition of the complexified tangent bundle TMC (which is the vector bundle of complexified tangent spaces at each point) into TM+ and TM-. A section of TM+ is called a vector field of type (1,0), while a section of TM- is an vector fie ...
